Electronic Communications in Probability

On Fixation of Activated Random Walks

Ori Gurel-Gurevich and Gideon Amir

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We prove that for the Activated Random Walks model on transitive unimodular graphs, if there is fixation, then every particle eventually fixates, almost surely. We deduce that the critical density is at most 1. Our methods apply for much more general processes on unimodular graphs. Roughly put, our result apply whenever the path of each particle has an automorphism invariant distribution and is independent of other particles' paths, and the interaction between particles is automorphism invariant and local. In particular, we do not require the particles path distribution to be Markovian. This allows us to answer a question of Rolla and Sidoravicius, in a more general setting then had been previously known (by Shellef).

Article information

Electron. Commun. Probab., Volume 15 (2010), paper no. 12, 119-123.

Accepted: 26 April 2010
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35]

Activated Random Walks Interacting Particles System

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Gurel-Gurevich, Ori; Amir, Gideon. On Fixation of Activated Random Walks. Electron. Commun. Probab. 15 (2010), paper no. 12, 119--123. doi:10.1214/ECP.v15-1536. https://projecteuclid.org/euclid.ecp/1465243955

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  • R. Dickman, L.T. Rolla, V. Sidoravicius Activated Random Walkers: Facts, Conjectures and Challenges Journal of Statistical Physics 138 (2010), 126-142.
  • R. Lyons, Y. Peres Probability on Trees and Networks Cambridge University Press
  • L.T. Rolla Generalized Hammersley Process and Phase Transition for Activated Random Walk Models
  • L.T. Rolla, V. Sidoravicius Absorbing-State Phase Transition for Stochastic Sandpiles and Activated Random Walks
  • E. Shellef, Nonfixation for Activated Random Walks